[Paper Review] Wilson operator algebras and ground states for coupled BF theories

Apoorv Tiwari, Xiao Chen+1 more|arXiv (Cornell University)|Mar 28, 2016

Algebraic structures and combinatorial models3 citations

TL;DR

This paper studies coupled BF theories in (3+1)D with cubic or quartic interactions, which support fractional braiding statistics among loop-like excitations. By canonically quantizing these theories on a three-torus, the authors derive the Wilson operator algebra, compute the S- and T-matrices encoding three- and four-loop braiding statistics, and construct ground state wave functions using geometric quantization with holomorphic and Hodge polarizations, revealing composite particles from Hopf links and Borromean rings of loops and point-like anyons.

ABSTRACT

The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theories on the three-torus, we explicitly calculate the $\mathcal{S}$- and $\mathcal{T}$-matrices, which encode fractional braiding statistics and topological spin of loop-like excitations, respectively. In the coupled $BF$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with point-like excitations, form composite particles.

Motivation & Objective

• To understand the algebra of Wilson loop and surface operators in coupled BF theories with cubic and quartic interactions in (3+1)D.
• To compute the S- and T-matrices that encode three-loop and four-loop braiding statistics and topological spins of loop excitations.
• To construct explicit ground state wave functions on the three-torus using geometric quantization with different polarizations.
• To identify composite particles formed by loop excitations (e.g., Hopf links, Borromean rings) and point-like anyons in these topological field theories.

Proposed method

• Canonical quantization of coupled BF theories with cubic and quartic terms on a three-torus spatial manifold.
• Mode decomposition of gauge fields and derivation of the zero-mode algebra from the symplectic structure of the theory.
• Construction of Wilson operator algebra using holomorphic and Hodge polarizations for the three-flavor quadratic and coupled BF theories.
• Explicit computation of the S-matrix and T-matrix from the Wilson operator algebra to extract fractional braiding statistics and topological spins.
• Use of large gauge invariance constraints to derive conditions on wave function coefficients, solved via theta functions and periodicity conditions.
• Two distinct wave function constructions: one using $\alpha^I_i$ and $\Lambda^I_i$ with holomorphic polarization, and another using $\alpha^I_i$ and $\beta^I_i$ with Hodge polarization.

Experimental results

Research questions

• RQ1 How do Wilson loop and surface operators algebraically relate in coupled BF theories with cubic and quartic couplings in (3+1)D?
• RQ2 What are the S- and T-matrices that encode the three-loop and four-loop braiding statistics in these theories?
• RQ3 How do large gauge transformations act on the Wilson operators and wave functions in the presence of cubic and quartic couplings?
• RQ4 What is the structure of the ground state wave function on the three-torus, and how does it depend on the choice of polarization?
• RQ5 How do composite particles formed by loop excitations (e.g., Hopf links, Borromean rings) emerge from the interplay of loop and point-like anyons in these theories?

Key findings

• The S-matrix and T-matrix for three-loop braiding are explicitly computed, revealing non-Abelian fractional statistics among three loop excitations in the cubic-coupled BF theory.
• The T-matrix encodes the topological spin of loop excitations, which is fractional and depends on the coupling constant $\mathrm{K}$ and the cubic coupling $\mathrm{K}\mathrm{r}$.
• Ground state wave functions are constructed in two ways: one using $\alpha^I_i$ and $\Lambda^I_i$ with holomorphic polarization, and another using $\alpha^I_i$ and $\beta^I_i$ with Hodge polarization, both satisfying large gauge invariance.
• The large gauge transformations act trivially on $\Lambda^I_i$ but non-trivially on $\beta^I_i$, leading to different constraints on the wave function coefficients.
• The wave function coefficients $C_I(p)$ satisfy a periodicity condition $C_I(p + \mathrm{K}m) = e^{i\theta} C_I(p)$, solvable via theta functions, ensuring gauge invariance.
• Composite particles arise from the linking of three loops (Hopf link) or four loops (Borromean ring), with anyons forming bound states that carry non-trivial braiding statistics.